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Keywords
(6)
Differential Forms
Harmonic Map
Mathematical Analysis
quasiconformal mapping
sobolev space
Variational Analysis
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nHarmonic mappings between annuli
nHarmonic mappings between annuli,Tadeusz Iwaniec,Jani Onninen
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nHarmonic mappings between annuli
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Citations: 4
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Tadeusz Iwaniec
,
Jani Onninen
The central theme of this paper is the
variational analysis
of homeomorphisms $h\colon \mathbb X \onto \mathbb Y$ between two given domains $\mathbb X, \mathbb Y \subset \mathbb R^n$. We look for the extremal mappings in the
Sobolev space
$\mathscr W^{1,n}(\mathbb X,\mathbb Y)$ which minimize the energy integral \[ \mathscr E_h=\int_{\mathbb X} Dh(x)^n dx. \] Because of the natural connections with quasiconformal mappings this $n$harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal $n$harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art.
Published in 2011.
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Citation Context
(2)
...We invoke the results in [1,
4
]. Within the Nitsche range (1.5) for the annuli A and A∗ the minimum is obtained (uniquely up to rotation) by the harmonic mapping...
...Outside the Nitsche range (1.5) for the annuli A and A∗ the infimum of H(A, A∗) is not attained [1,
4
]. It is a general fact concerning mappings...
...The reader may wish to know that the nonharmonic mapping h(z) = z/z of A onto the unit circle is a minimizer of the Dirichlet integral [
4
]...
Tadeusz Iwaniec
,
et al.
Harmonic mappings of an annulus, Nitsche conjecture and its generaliza...
...integrals. Let us now look at the example in which the existence of deformations of finite conformal energy is lacking [
22
]...
...Note that J(z,h) 0 for r 6 z 6 . This minimizer is actually unique up to the rotation of annuli, [
22
]...
...! Y whose integral depends only on the homotopy class of h, regardless of its boundary values, [
22
, 23, 3]. The key to the proof of Theorem 1 is a sharp pointwise estimate of the storedenergy function by means of free Lagrangians, say Dh(x)2 + J(x,h) > L(x,h,Dh) (16)...
...This is not always the case, and we have some surprise for the reader [
22
]...
Tadeusz Iwaniec
,
et al.
Neohookean Deformations of Annuli
References
(37)
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(
Citations: 7
)
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,
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,
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(
Citations: 26
)
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,
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Journal:
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Published in 1977.
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(
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B. Bojarski
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Published in 1983.
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Citations
(4)
Deformations of finite conformal energy: existence and removability of singularities
(
Citations: 3
)
Tadeusz Iwaniec
,
Jani Onninen
Journal:
Proceedings of The London Mathematical Society  PROC LONDON MATH SOC
, vol. 100, no. 1, pp. 123, 2010
Harmonic mappings of an annulus, Nitsche conjecture and its generalizations
(
Citations: 4
)
Tadeusz Iwaniec
,
Leonid V. Kovalev
,
Jani Onninen
Published in 2009.
Neohookean Deformations of Annuli
Tadeusz Iwaniec
,
Jani Onninen
Cavitation is not Allowed
Tadeusz Iwaniec
,
Jani Onninen